[iseswemp] Orbiting dust

Consider a collection of particles scattered with approximate uniform density in a sphere.  Each particle travels in a circular orbit around a central point mass.  The particles do not influence each other gravitationally.  This is easy to set up in simulation.  For each particle, pick an orbital inclination uniformly randomly.

Next, consider a collection of particles all with the same speed but traveling in uniformly random directions.  Easiest first is probably to consider a spherical shell of such particles.  Place a point mass at the center and let gravity do its work.  Again, none of the particles interact with each other so this is a simple 2-body problem repeated for each particle.  Some particles may escape.  For a given initial velocity and shell radius, what is the spatial distribution of (non-escaped) particles averaged over time, as a function of distance from the central point?  Add some more shells to get a somewhat uniform density collection of particles (within a certain radius) buzzing around the central mass in elliptical orbits.

The point of all of this is then to add another massive body to make it a 3-body problem.  Dust will get ejected from certain regions, collect in other regions (Lagrange points).

Useful are symplectic integrators, e.g., Verlet.